Motion in 2-dimensions Using Vector Notation Mark Lesmeister AP Physics C Dawson High School.
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Transcript of Motion in 2-dimensions Using Vector Notation Mark Lesmeister AP Physics C Dawson High School.
Motion in 2-dimensions Using Vector Notation
Mark LesmeisterAP Physics C
Dawson High School
© Mark Lesmeister/Pearland ISDSelected graphics from Tipler, “Physics for Scientists and Engineers,”, © 2008 W.H. Freeman and Company.Selected questions taken from Young and Freedman, “Instructor’s Resource DVD for University Physics”, © 2012 Pearson Education Inc.Selected questions taken from Cutnell and Johnson, “Instructor Companion Site for Physics, 9th Edition”, © 2014 John Wiley and Sons
Acknowledgements
KINEMATICS WITH VECTOR NOTATION
Section 1
Position Vectors The position
vector is the vector from the origin to the position of the object.
jiryx
(x,y)r
x
y
Position and Displacement Vectors
The displacement vector is the difference between two position vectors.
12 rrr jir
)()( 1212 yyxx
1r
2r
r
Average Velocity
The average velocity vector is the ratio of the displacement vector to the time interval.
Since the time interval is a scalar, this vector has the same direction as the displacement.
tΔr
vAV
Average Velocity Example
velocity. theof magnitude thefind
Then form.component in stated interval, for this velocity average
theFind . m220 m 40 isposition its , at time later,
minutes Two . m 100 m 400 ofposition a hassailboat A
2 jir
jir
2
1
t
jir
vAV
t
y
t
x
t
jivAV
s 120
m )100220(
s 120
m )40040(
jivAV m/s 0.1 m/s 0.3
Instantaneous velocity
The instantaneous velocity vector is the derivative of the position vector with respect to time.
jir
v
dt
dy
dt
dx
dt
d
jivyx vv
Instantaneous velocity example
vector. velocity ousinstantane theFind
.1800200)( and 2.0100)(by given position a hassailboat A 1 ttyttx
jir
v
dt
dy
dt
dx
dt
d
jiv
218002.0 t
Constant Velocity Model in Two Dimensions
jivr
jiv
tvtvtt
vvvv
yx
yxyx
)()(
.directionsy and x in the elocitiesconstant v theare
and where
Constant Acceleration Model in 2 Dimensions
ravvvv
avv
avrr
00
0
00
2
221
t
tt
Constant Acceleration Model in 2 Dimensions x-direction y-direction
xavv
tavtv
tatvxtx
xxx
xxx
xx
2
)(
)(
20
2
0
221
00
yavv
tavtv
tatvyty
yyy
yyy
yy
2
)(
)(
20
2
0
221
00
RELATIVE VELOCITYSection 2
Relative Velocity A physics student is riding on a light rail
train travelling at 20 mi/hr. The student walks toward the front of the train with a speed of 5 mi/hr in the direction of the train’s motion. How fast does the student appear to be moving relative to the ground?
Relative Velocity Let vAB be the velocity of object A relative to object B. Let vBC be the velocity of object B relative to object C. Then vAC, the velocity of object A relative to object C,
is given by
vAC = vAB + vBC
Relative Velocity Practice
http://cnx.org/content/m42045/latest/?collection=col11406/1.7
PROJECTILE MOTIONSection 3
Projectile motion
The x and y motions are independent of each other. In the x-direction, the motion is
constant velocity. In the y direction, the motion is
constant acceleration, with a =-9.8 m/s2
jir
)()()( tytxt
jiv
)()( tvtv yx
Equations of motion for a projectile x-direction y-direction
0
)(
)(
0
221
00
x
xxx
xx
a
tavtv
tatvxtx
xx
x
vtv
tvxtx
0
00
)(
)(
ga
tavtv
tatvyty
y
yyy
yy
0
221
00
)(
)(
gtvtv
gttvyty
yy
y
0
221
00
)(
)(
©2008 by W.H. Freeman and Company
©2008 by W.H. Freeman and Company
Projectile Motion Example
A physics student throws his cap into the air with an initial velocity of 24.5 m/s at 36.9o from the horizontal. Find the total time the cap is in the air, and the total distance travelled.
©2008 by W.H. Freeman and Company
Projectile Motion Practice
http://cnx.org/content/m42042/latest/?collection=col11406/1.7